Courbures, groupes de Galois génériques et D-groupoïde de Galois d'un système aux q-différences

Abstract

Combining the results in Hendriks (1996) [6], Di Vizio (2002) [1] and Di Vizio, Hardouin [2], we prove that the generic, algebraic or differential, Galois group of a q-difference modules over C ( x ) can always be characterized in terms of v-curvatures, in the spirit of the work of Katz (1982) [8]. We use this result to prove that the Malgrange–Granier D-groupoid of a linear q-difference system coincide, in a sense that we specify below, with a sort of Kolchin closure of the dynamics of the linear q-difference system and that the group that fixes a transversal, coincide with the differential generic Galois group.